FINC621 Winter 2018-19 Extra Credit (10pts)

Asian Option Pricing & MC Simulation

About

This assignment is an extra credit opportunity worth 10 points. The goal is to demonstrate your understanding of the topic in the different areas of problem statement through a flowchart diagram propely marked, the analytics through a clear representation of the mathematical equations, the coding through the clear and properly commented execution steps, and finally results evaluation through the assessment of quality and integrity.

The topic of interest is the pricing of an Asian option. Asian options are a form of exotic options unlike the more vanilla European options. The case considered here is a special type of Asian options where the payoff is given by the average price of the underline asset over some predetermined period until expiration time. Other variations of Asian options (outside our scope) are possible and left for the reader’s investigation. The case in point is described on p.169 . Note that Asian options are characterized as path dependent options. By this we mean that the payoff of the option is dependent on the intermediate values of the underline asset along the path. In contrast a European option is path independent because the payoff depends only on the final value of the underline asset, and not on the intermediate or historical values. The case of American options is slightly tricky. Because of the early exercise opportunity one may be inclined to view American options as path dependendent. However at any exercise point the payoff is only dependent on the current asset price and not historical. As such it is not really path dependent.

Setup

Remember to always set your working directory to the source file location. Go to ‘Session’, scroll down to ‘Set Working Directory’, and click ‘To Source File Location’. Read carefully the below and follow the instructions to complete the tasks and answer any questions. Submit your work to RPubs as detailed in previous notes.

Note

Always read carefully the instructions on Sakai. For clarity, tasks/questions to be completed/answered are highlighted in red color (visible in preview) and numbered according to their particular placement in the task section. Quite often you will need to add your own code chunk.

Execute all code chunks, preview, publish, and submit link on Sakai follwoing the naming convention. Make sure to add comments to your code where appropriate. Use own language!

Any sign of plagiarism, will result in dissmissal of work!

---

#Install package quantmod 
if(!require("quantmod",quietly = TRUE))
  install.packages("quantmod",dependencies = TRUE, repos = "https://cloud.r-project.org")
if(!require("fOptions",quietly = TRUE))
  install.packages("fOptions",dependencies = TRUE, repos = "https://cloud.r-project.org")

Attaching package: <U+393C><U+3E31>timeSeries<U+393C><U+3E32>

The following object is masked from <U+393C><U+3E31>package:zoo<U+393C><U+3E32>:

    time<-


Attaching package: <U+393C><U+3E31>fBasics<U+393C><U+3E32>

The following object is masked from <U+393C><U+3E31>package:TTR<U+393C><U+3E32>:

    volatility

Question 1 : Flowchart Diagram (2pts)

Inspired by the flowchart diagram of the MC simulation for European option, create an equivalent flowchart diagram for the considered case of Asian option. Describe the variables, label clearly, and write the discrete mathematical representation to be coded later. You can create the flowchart using a separate tool and include an image capture here. A hand drawn image is acceptable only if well executed.

Call= max((1/n)*Asum-K,0)

Put= max(K-(1/n)*Asum,0)

Where Asum is the average of all the share prices throughout the simulation.

K is the strike price and n is the number of periods

\(PV_{Call or Put}\)= Average option price *\(e^{-rf*t}\)

\(S_{t+\Delta t}\)=\(S_{t}\)+r\(S_{t}\)\(\Delta\) t + \(\sigma\)\(S_{t}\)\(\sqrt{\Delta t}\)*Rnd

where- Rnd called E in the code is a random number from standard normal distribution.

\(\Delta\) t = T/n

Question 2: MC Simulation (2pts)

Implement the R-code for the MC simulation to price an Asian Call option with input characteristics similar to the European Call option considered in Lab 6, task 2B.

Initial price of $155, strike price $140, a time to expiration equal to six months, a risk-free interest rate of 2.5% and a volatility of 23%. Consider a number of periods n=100 and a number of simulations m=1000

MCAsian= function(Type="c",S=155,K=140,t=1/2,r=0.025,sigma=0.23,n=100,m=1000,dt=t/n){
  baseS=S
  sum=0;
  for(i in 1:m) {#number of simulation paths}
    S=baseS
    Asum=0
    for(j in 1:n){
    E=rnorm(1,0,1);
    S=S+r*S*dt+sigma*S*sqrt(dt)*E;
    Asum=Asum+S}
    
  if(Type=="c"){payoff=max((1/n)*Asum-K,0)}
  else if (Type=="p"){payoff=max(K-(1/n)*Asum,0)}
  else{payoff=max((1/n)*Asum-K,0)} #default
  sum=sum+payoff}
OptionValue=(sum*exp(-r*t))/m;
return(OptionValue)}
x=MCAsian(Type="c",S=155,K=140,t=1/2,r=0.025,sigma=0.23,n=100,m=1000,dt=0.5/100)
x

[1] 16.81855

Question 3: Function Calculation (2pts)

For this question the below package is needed. Follow the example at bottom of p.171 to price the same Asian Call option using instead the approximate function referenced in the example. Compare to the MC simulation.

Solution- In our findings, there isn’t much difference in the option price obtained from both the methods because the payoff in asian options is averaged, this takes out much of the difference that is created by the MC simulation.

 TW = TurnbullWakemanAsianApproxOption(
                     TypeFlag = "c", S = 155, SA = 140, X = 140, 
                     Time = 1/2, time = 1/2, tau = 0 , r = 0.025,
                     b = r, sigma = 0.23)
      print (TW)

Title:
 Turnbull Wakeman Asian Approximated Option 

Call:
 TurnbullWakemanAsianApproxOption(TypeFlag = "c", S = 155, SA = 140, 
     X = 140, Time = 1/2, time = 1/2, tau = 0, r = 0.025, b = r, 
     sigma = 0.23)

Parameters:
          Value:
 TypeFlag c     
 S        155   
 SA       140   
 X        140   
 Time     0.5   
 time     0.5   
 tau      0     
 r        0.025 
 b        0.025 
 sigma    0.23  

Option Price:
 16.63123 

Description:
 Tue Feb 05 00:09:21 2019 

Question 4: Greeks Calculation (2pts)

Calculating the Greeks for the Asian Option is not as straightforward using the approximate function. Instead we can approximate using numerical differentiation. Describe how would you implement a numerical differentiation to calculate the Delta. Show the code execution and the intermediate values to calculate the Delta and the Theta for the considered Asian Call option.

Initial price of $155, strike price $140, a time to expiration equal to six months, a risk-free interest rate of 2.5% and a volatility of 23%.

The Delta is 0.878185

 TW1 = TurnbullWakemanAsianApproxOption(
                     TypeFlag = "c", S = 154, SA = 140, X = 140, 
                     Time = 1/2, time = 1/2, tau = 0 , r = 0.025,
                     b = r, sigma = 0.23)
 TW2 = TurnbullWakemanAsianApproxOption(
                     TypeFlag = "c", S = 156, SA = 140, X = 140, 
                     Time = 1/2, time = 1/2, tau = 0 , r = 0.025,
                     b = r, sigma = 0.23)
 DELTA= (17.51607-15.7597)/2
 print(DELTA)

[1] 0.878185

Considering change in time(dt) as 1/12, we get the Theta as 4.74656

  TW3 = TurnbullWakemanAsianApproxOption(
                     TypeFlag = "c", S = 155, SA = 140, X = 140, 
                     Time = 7/12, time = 1/2, tau = 0 , r = 0.025,
                     b = r, sigma = 0.23)
 TW4 = TurnbullWakemanAsianApproxOption(
                     TypeFlag = "c", S = 155, SA = 140, X = 140, 
                     Time = 5/12, time = 1/2, tau = 0 , r = 0.025,
                     b = r, sigma = 0.23)
 
 Theta= (16.62982 -14.25654 )/0.50
 print(Theta)

[1] 4.74656

Question 5: Evaluation of Results (2pts)

Explain how the price of the Asian Call option compares to the equivalent European Call option. Is it an expected behavior? What about the Delta comparison? Poorly stated insights will receive poor grade!

We notice that Asian call option price are comparatively cheaper from it’s european counterpart which are priced in the range of 18 to 22 (lab6). This is caused by the averaging effect in the asian options which reduce the volatility, which in turn reduce the price.

When we look at the Delta prices for European option which is 0.78341 and the asian option which is 0.878185 we can conclude that asian options are comparatively more sensitive to the change in the underlying price, which is a prominent feature of asian options(their payoff depends on them)

*http://computationalfinance.lsi.upc.edu